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Primitive norm extraction and a signed descent for the cubic Fermat equation
Abstract
We give a self-contained proof of the cubic case of Fermat's theorem through a signed form of Euler's descent, with the required extraction step proved directly in ℤ[√(−3)]. A pigeonhole argument of Thue represents the relevant primes by X2+3Y2, and explicit division in ℤ[√(−3)] lifts primitive odd-power norms to powers in that order. For cubes, the resulting coordinate identities transform 2K(K2+3M2)=C3 into another equation of the same signed form with a strictly smaller non-zero integer height. The construction gives |C'|≤ |C|/39. Half-sums and half-differences of the two odd entries place the sum and difference parity configurations in this single descent.
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BibTeX
@misc{laniewski2026primitive,
title = {{Primitive norm extraction and a signed descent for the cubic Fermat equation}},
author = {R. Laniewski},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28566},
eprint = {2609.28566},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28566},
}
RIS
TY - UNPB
TI - Primitive norm extraction and a signed descent for the cubic Fermat equation
AU - R. Laniewski
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.28566
UR - https://arxiv.org/abs/2609.28566
AB - We give a self-contained proof of the cubic case of Fermat's theorem through a signed form of Euler's descent, with the required extraction step proved directly in ℤ[√(−3)]. A pigeonhole argument of Thue represents the relevant primes by X^2+3Y^2, and explicit division in ℤ[√(−3)] lifts primitive odd-power norms to powers in that order. For cubes, the resulting coordinate identities transform 2K(K^2+3M^2)=C^3 into another equation of the same signed form with a strictly smaller non-zero integer height. The construction gives |C'|≤ |C|/39. Half-sums and half-differences of the two odd entries place the sum and difference parity configurations in this single descent.
ER -